atom


Let P be a poset, partially ordered by . An element aP is called an atom if it covers some minimal element of P. As a result, an atom is never minimalPlanetmathPlanetmath. A poset P is called atomic if for every element pP that is not minimal has an atom a such that ap.

Examples.

  1. 1.

    Let A be a set and P=2A its power setMathworldPlanetmath. P is a poset ordered by with a unique minimal element . Thus, all singleton subsets {a} of A are atoms in P.

  2. 2.

    + is partially ordered if we define ab to mean that ab. Then 1 is a minimal element and any prime number p is an atom.

Remark. Given a latticeMathworldPlanetmath L with underlying poset P, an element aL is called an atom (of L) if it is an atom in P. A lattice is a called an atomic lattice if its underlying poset is atomic. An atomistic lattice is an atomic lattice such that each element that is not minimal is a join of atoms. If a is an atom in a semimodular lattice L, and if a is not under x, then ax is an atom in any interval lattice I where x=I.

Examples.

  1. 1.

    P=2A, with the usual intersectionMathworldPlanetmath and union as the lattice operationsMathworldPlanetmath meet and join, is atomistic: every subset B of A is the union of all the singleton subsets of B.

  2. 2.

    +, partially ordered as above, with lattice binary operationsMathworldPlanetmath defined by ab=gcd(a,b), and ab=lcm(a,b), is a lattice that is atomic, as we have seen earlier. But it is not atomistic: 4 is not a join of 2’s; 36 is not a join of 2 and 3 are just two counterexamples.

Title atom
Canonical name Atom
Date of creation 2013-03-22 15:20:09
Last modified on 2013-03-22 15:20:09
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 13
Author CWoo (3771)
Entry type Definition
Classification msc 06A06
Classification msc 06B99
Defines atomic poset
Defines atomic lattice
Defines atomistic lattice
Defines atomistic